then as n goes to infinity, Xn converges in probability (see below) to the common mean, μ, of the random variables Yi. This result is known as the weak law of large numbers. Other forms of convergence are important in other useful theorems, including the central limit theorem.

Throughout the following, we assume that (Xn) is a sequence of random variables, and X is a random variable, and all of them are defined on the same probability space (Ω, F, P).

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Suppose that F1, F2, ... is a sequence of cumulative distribution functions corresponding to random variables X1, X2, ..., and that F is a distribution function corresponding to a random variable X. We say that the sequence Xn converges towards Xin distribution, if

for every real numbera at which F is continuous. Since F(a) = Pr(X ≤ a), this means that the probability that the value of X is in a given range is very similar to the probability that the value of Xn is in that range, provided n is large enough. Convergence in distribution is often denoted by adding the letter over an arrow indicating convergence:

Convergence in distribution is the weakest form of convergence, and is sometimes called weak convergence. It does not, in general, imply any other mode of convergence. However, convergence in distribution is implied by all other modes of convergence mentioned in this article, and hence, it is the most common and often the most useful form of convergence of random variables. It is the notion of convergence used in the central limit theorem and the (weak) law of large numbers.

A useful result, which may be employed in conjunction with law of large numbers and the central limit theorem, is that if a function g: R → R is continuous, then if Xn converges in distribution to X, then so too does g(Xn) converge in distribution to g(X). (This may be proved using Skorokhod's representation theorem.)

Convergence in distribution is also called convergence in law, since the word "law" is sometimes used as a synonym of "probability distribution."

for every ε > 0. Convergence in probability is, indeed, the (pointwise) convergence of probabilities. Pick any ε > 0 and any δ > 0. Let Pn be the probability that Xn is outside a tolerance ε of X. Then, if Xn converges in probability to X then there exists a value N such that, for all n ≥ N, Pn is itself less than δ.

Convergence in probability is often denoted by adding the letter 'P' over an arrow indicating convergence:

Convergence in probability is the notion of convergence used in the weak law of large numbers.
Convergence in probability implies convergence in distribution. To prove it, it's convenient to prove the following, simple lemma:

We say that the sequence Xn converges almost surely or almost everywhere or with probability 1 or strongly towards X if

This means that you are virtually guaranteed that the values of Xn approach the value of X, in the sense (see almost surely) that events for which Xn does not converge to X have probability 0. Using the probability space (Ω, F, P) and the concept of the random variable as a function from Ω to R, this is equivalent to the statement

Almost sure convergence implies convergence in probability, and hence implies convergence in distribution. It is the notion of convergence used in the strong law of large numbers.

The chain of implications between the various notions of convergence, above, are noted in their respective sections, but it is sometimes important to establish converses to these implications. No other implications other than those noted above hold in general, but a number of special cases do permit converses:

If Xn converges in distribution to a constant c, then Xn converges in probability to c.

If Xn converges in probability X, and if Pr(|Xn| ≤ b) = 1 for all n and some b, then Xn converges in rth mean to X for all r ≥ 1. In other words, if Xn converges in probability to X and all random variables Xn are almost surely bounded above and below, then Xn converges to X also in any rth mean.

If for all ε > 0,

then Xn converges almost surely to X. In other words, if Xn converges in probability to X sufficiently quickly (i.e. the above sum converges for all ε > 0), then Xn also converges almost surely to X.

If Sn is a sum of n real independent random variables:

then Sn converges almost surely if and only if Sn converges in probability.